A.10.5: Section 10.5 Answers
- Page ID
- 43738
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)1. \({\bf y}=c_{1}\left[\begin{array}{c}{2}\\[4pt]{1}\end{array}\right]e^{5t}+c_{2}\left(\left[\begin{array}{c}{-1}\\[4pt]{0}\end{array}\right]e^{5t}+\left[\begin{array}{c}{2}\\[4pt]{1}\end{array}\right]te^{5t} \right)\)
2. \({\bf y}=c_{1}\left[\begin{array}{c}{1}\\[4pt]{1}\end{array}\right]e^{-t}+c_{2}\left(\left[\begin{array}{c}{1}\\[4pt]{0}\end{array}\right]e^{-t}+\left[\begin{array}{c}{1}\\[4pt]{1}\end{array}\right]te^{-t} \right)\)
3. \({\bf y}=c_{1}\left[\begin{array}{c}{-2}\\[4pt]{1}\end{array}\right]e^{-9t}+c_{2}\left(\left[\begin{array}{c}{-1}\\[4pt]{0}\end{array}\right]e^{-9t}+\left[\begin{array}{c}{-2}\\[4pt]{1}\end{array}\right]te^{-9t} \right)\)
4. \({\bf y}=c_{1}\left[\begin{array}{c}{-1}\\[4pt]{1}\end{array}\right]e^{2t}+c_{2}\left(\left[\begin{array}{c}{-1}\\[4pt]{0}\end{array}\right]e^{2t}+\left[\begin{array}{c}{-1}\\[4pt]{1}\end{array}\right]te^{2t} \right)\)
5. \({\bf y}=c_{1}\left[\begin{array}{c}{-1}\\[4pt]{1}\end{array}\right]+c_{2}\left(\left[\begin{array}{c}{-1}\\[4pt]{0}\end{array}\right]\frac{e^{-2t}}{3}+\left[\begin{array}{c}{-2}\\[4pt]{1}\end{array}\right]te^{-2t} \right)\)
6. \({\bf y}=c_{1}\left[\begin{array}{c}{3}\\[4pt]{2}\end{array}\right]e^{-4t}+c_{2}\left(\left[\begin{array}{c}{-1}\\[4pt]{0}\end{array}\right]\frac{e^{-4t}}{2}+\left[\begin{array}{c}{3}\\[4pt]{2}\end{array}\right]te^{-4t} \right)\)
7. \({\bf y}=c_{1}\left[\begin{array}{c}{4}\\[4pt]{3}\end{array}\right]e^{-t}+c_{2}\left(\left[\begin{array}{c}{-1}\\[4pt]{0}\end{array}\right]\frac{e^{-t}}{3}+\left[\begin{array}{c}{4}\\[4pt]{3}\end{array}\right]te^{-t} \right)\)
8. \({\bf y}=c_{1}\left[\begin{array}{c}{-1}\\[4pt]{-1}\\[4pt]{2}\end{array}\right]+c_{2}\left[\begin{array}{c}{1}\\[4pt]{1}\\[4pt]{2}\end{array}\right]e^{4t}+c_{3}\left(\left[\begin{array}{c}{0}\\[4pt]{1}\\[4pt]{0}\end{array}\right]\frac{e^{4t}}{2}+\left[\begin{array}{c}{1}\\[4pt]{1}\\[4pt]{2}\end{array}\right]te^{4t} \right)\)
9. \({\bf y}=c_{1}\left[\begin{array}{c}{-1}\\[4pt]{1}\\[4pt]{1}\end{array}\right]e^{t}+c_{2}\left[\begin{array}{c}{1}\\[4pt]{-1}\\[4pt]{1}\end{array}\right]e^{-t}+c_{3}\left(\left[\begin{array}{c}{0}\\[4pt]{3}\\[4pt]{0}\end{array}\right]e^{-t}+\left[\begin{array}{c}{1}\\[4pt]{-1}\\[4pt]{1}\end{array}\right]te^{-t} \right)\)
10. \({\bf y}=c_{1}\left[\begin{array}{c}{0}\\[4pt]{1}\\[4pt]{1}\end{array}\right]e^{2t}+c_{2}\left[\begin{array}{c}{1}\\[4pt]{0}\\[4pt]{1}\end{array}\right]e^{-2t}+c_{3}\left(\left[\begin{array}{c}{1}\\[4pt]{1}\\[4pt]{0}\end{array}\right]\frac{e^{-2t}}{2}+\left[\begin{array}{c}{1}\\[4pt]{0}\\[4pt]{1}\end{array}\right]te^{-2t} \right)\)
11. \({\bf y}=c_{1}\left[\begin{array}{c}{-2}\\[4pt]{-3}\\[4pt]{1}\end{array}\right]e^{2t}+c_{2}\left[\begin{array}{c}{0}\\[4pt]{-1}\\[4pt]{1}\end{array}\right]e^{4t}+c_{3}\left(\left[\begin{array}{c}{1}\\[4pt]{0}\\[4pt]{0}\end{array}\right]\frac{e^{4t}}{2}+\left[\begin{array}{c}{0}\\[4pt]{-1}\\[4pt]{1}\end{array}\right]te^{4t} \right)\)
12. \({\bf y}=c_{1}\left[\begin{array}{c}{-1}\\[4pt]{-1}\\[4pt]{1}\end{array}\right]e^{-2t}+c_{2}\left[\begin{array}{c}{1}\\[4pt]{1}\\[4pt]{1}\end{array}\right]e^{4t}+c_{3}\left(\left[\begin{array}{c}{1}\\[4pt]{0}\\[4pt]{0}\end{array}\right]\frac{e^{4t}}{2}+\left[\begin{array}{c}{1}\\[4pt]{1}\\[4pt]{1}\end{array}\right]te^{4t} \right)\)
13. \({\bf y}=\left[\begin{array}{c}{6}\\[4pt]{2}\end{array}\right]e^{-7t}- \left[\begin{array}{c}{8}\\[4pt]{4}\end{array}\right]te^{-7t}\)
14. \({\bf y}=\left[\begin{array}{c}{5}\\[4pt]{8}\end{array}\right]e^{3t}- \left[\begin{array}{c}{12}\\[4pt]{16}\end{array}\right]te^{3t}\)
15. \({\bf y}=\left[\begin{array}{c}{2}\\[4pt]{3}\end{array}\right]e^{-5t}- \left[\begin{array}{c}{8}\\[4pt]{4}\end{array}\right]te^{-5t}\)
16. \({\bf y}=\left[\begin{array}{c}{3}\\[4pt]{1}\end{array}\right]e^{5t}- \left[\begin{array}{c}{12}\\[4pt]{6}\end{array}\right]te^{5t}\)
17. \({\bf y}=\left[\begin{array}{c}{0}\\[4pt]{2}\end{array}\right]e^{-4t}+ \left[\begin{array}{c}{6}\\[4pt]{6}\end{array}\right]te^{-4t}\)
18. \({\bf y}=\left[\begin{array}{c}{4}\\[4pt]{8}\\[4pt]{-6}\end{array}\right]e^{t}+ \left[\begin{array}{c}{2}\\[4pt]{-3}\\[4pt]{-1}\end{array}\right]e^{-2t}+\left[\begin{array}{c}{-1}\\[4pt]{1}\\[4pt]{0}\end{array}\right]te^{-2t}\)
19. \({\bf y}=\left[\begin{array}{c}{3}\\[4pt]{3}\\[4pt]{6}\end{array}\right]e^{2t}- \left[\begin{array}{c}{9}\\[4pt]{5}\\[4pt]{6}\end{array}\right]+\left[\begin{array}{c}{2}\\[4pt]{2}\\[4pt]{0}\end{array}\right]t\)
20. \({\bf y}=-\left[\begin{array}{c}{2}\\[4pt]{0}\\[4pt]{2}\end{array}\right]e^{-3t}+ \left[\begin{array}{c}{-4}\\[4pt]{9}\\[4pt]{1}\end{array}\right]e^{t}-\left[\begin{array}{c}{0}\\[4pt]{4}\\[4pt]{4}\end{array}\right]te^{t}\)
21. \({\bf y}=\left[\begin{array}{c}{-2}\\[4pt]{2}\\[4pt]{2}\end{array}\right]e^{4t}+ \left[\begin{array}{c}{0}\\[4pt]{-1}\\[4pt]{1}\end{array}\right]e^{2t}+\left[\begin{array}{c}{3}\\[4pt]{-3}\\[4pt]{3}\end{array}\right]te^{2t}\)
22. \({\bf y}=-\left[\begin{array}{c}{1}\\[4pt]{1}\\[4pt]{0}\end{array}\right]e^{-4t}+ \left[\begin{array}{c}{-3}\\[4pt]{2}\\[4pt]{-3}\end{array}\right]e^{8t}+\left[\begin{array}{c}{8}\\[4pt]{0}\\[4pt]{-8}\end{array}\right]te^{8t}\)
23. \({\bf y}=\left[\begin{array}{c}{3}\\[4pt]{6}\\[4pt]{3}\end{array}\right]e^{4t}- \left[\begin{array}{c}{3}\\[4pt]{4}\\[4pt]{1}\end{array}\right]+\left[\begin{array}{c}{8}\\[4pt]{4}\\[4pt]{4}\end{array}\right]t\)
24. \({\bf y}=c_{1}\left[\begin{array}{c}{0}\\[4pt]{1}\\[4pt]{1}\end{array}\right]e^{6t} + c_{2}\left(\left[\begin{array}{c}{-1}\\[4pt]{1}\\[4pt]{0}\end{array}\right]\frac{e^{6t}}{4}+\left[\begin{array}{c}{0}\\[4pt]{1}\\[4pt]{1}\end{array}\right]te^{6t} \right)+c_{3}\left(\left[\begin{array}{c}{1}\\[4pt]{1}\\[4pt]{0}\end{array}\right]\frac{e^{6t}}{8}+\left[\begin{array}{c}{-1}\\[4pt]{1}\\[4pt]{0}\end{array}\right]\frac{te^{6t}}{4}+\left[\begin{array}{c}{0}\\[4pt]{1}\\[4pt]{1}\end{array}\right]\frac{t^{2}e^{6t}}{2} \right)\)
25. \({\bf y}=c_{1}\left[\begin{array}{c}{-1}\\[4pt]{1}\\[4pt]{1}\end{array}\right]e^{3t} + c_{2}\left(\left[\begin{array}{c}{1}\\[4pt]{0}\\[4pt]{0}\end{array}\right]\frac{e^{3t}}{2}+\left[\begin{array}{c}{-1}\\[4pt]{1}\\[4pt]{1}\end{array}\right]te^{3t} \right)+c_{3}\left(\left[\begin{array}{c}{1}\\[4pt]{2}\\[4pt]{0}\end{array}\right]\frac{e^{3t}}{36}+\left[\begin{array}{c}{1}\\[4pt]{0}\\[4pt]{0}\end{array}\right]\frac{te^{3t}}{2}+\left[\begin{array}{c}{-1}\\[4pt]{1}\\[4pt]{1}\end{array}\right]\frac{t^{2}e^{3t}}{2} \right)\)
26. \({\bf y}=c_{1}\left[\begin{array}{c}{0}\\[4pt]{-1}\\[4pt]{1}\end{array}\right]e^{-2t} + c_{2}\left(\left[\begin{array}{c}{-1}\\[4pt]{1}\\[4pt]{0}\end{array}\right]e^{-2t}+\left[\begin{array}{c}{0}\\[4pt]{-1}\\[4pt]{1}\end{array}\right]te^{-2t} \right)+c_{3}\left(\left[\begin{array}{c}{3}\\[4pt]{-2}\\[4pt]{0}\end{array}\right]\frac{e^{-2t}}{4}+\left[\begin{array}{c}{-1}\\[4pt]{1}\\[4pt]{0}\end{array}\right]te^{-2t}+\left[\begin{array}{c}{0}\\[4pt]{-1}\\[4pt]{1}\end{array}\right]\frac{t^{2}e^{-2t}}{2} \right)\)
27. \({\bf y}=c_{1}\left[\begin{array}{c}{0}\\[4pt]{1}\\[4pt]{1}\end{array}\right]e^{2t} + c_{2}\left(\left[\begin{array}{c}{1}\\[4pt]{1}\\[4pt]{0}\end{array}\right]\frac{e^{2t}}{2}+\left[\begin{array}{c}{0}\\[4pt]{1}\\[4pt]{1}\end{array}\right]te^{2t} \right)+c_{3}\left(\left[\begin{array}{c}{-1}\\[4pt]{1}\\[4pt]{0}\end{array}\right]\frac{e^{2t}}{8}+\left[\begin{array}{c}{1}\\[4pt]{1}\\[4pt]{0}\end{array}\right]\frac{te^{2t}}{2}+\left[\begin{array}{c}{0}\\[4pt]{1}\\[4pt]{1}\end{array}\right]\frac{t^{2}e^{2t}}{2} \right)\)
28. \({\bf y}=c_{1}\left[\begin{array}{c}{-2}\\[4pt]{1}\\[4pt]{2}\end{array}\right]e^{-6t} + c_{2}\left(-\left[\begin{array}{c}{6}\\[4pt]{1}\\[4pt]{0}\end{array}\right]\frac{e^{-6t}}{6}+\left[\begin{array}{c}{-2}\\[4pt]{1}\\[4pt]{2}\end{array}\right]te^{-6t} \right)+c_{3}\left(-\left[\begin{array}{c}{12}\\[4pt]{1}\\[4pt]{0}\end{array}\right]\frac{e^{-6t}}{36}-\left[\begin{array}{c}{6}\\[4pt]{1}\\[4pt]{0}\end{array}\right]\frac{te^{-6t}}{6}+\left[\begin{array}{c}{-2}\\[4pt]{1}\\[4pt]{2}\end{array}\right]\frac{t^{2}e^{-6t}}{2} \right)\)
29. \({\bf y}=c_{1}\left[\begin{array}{c}{-4}\\[4pt]{0}\\[4pt]{1}\end{array}\right]e^{-3t}+c_{2}\left[\begin{array}{c}{6}\\[4pt]{1}\\[4pt]{0}\end{array}\right]e^{-3t}+c_{3}\left(\left[\begin{array}{c}{1}\\[4pt]{1}\\[4pt]{0}\end{array}\right]e^{-3t}+\left[\begin{array}{c}{2}\\[4pt]{1}\\[4pt]{1}\end{array}\right]te^{-3t} \right)\)
30. \({\bf y}=c_{1}\left[\begin{array}{c}{-1}\\[4pt]{0}\\[4pt]{1}\end{array}\right]e^{-3t}+c_{2}\left[\begin{array}{c}{0}\\[4pt]{1}\\[4pt]{0}\end{array}\right]e^{-3t}+c_{3}\left(\left[\begin{array}{c}{1}\\[4pt]{1}\\[4pt]{0}\end{array}\right]e^{-3t}+\left[\begin{array}{c}{-1}\\[4pt]{-1}\\[4pt]{1}\end{array}\right]te^{-3t} \right)\)
31. \({\bf y}=c_{1}\left[\begin{array}{c}{2}\\[4pt]{0}\\[4pt]{1}\end{array}\right]e^{-t}+c_{2}\left[\begin{array}{c}{-3}\\[4pt]{2}\\[4pt]{0}\end{array}\right]e^{-t}+c_{3}\left(\left[\begin{array}{c}{1}\\[4pt]{0}\\[4pt]{0}\end{array}\right]\frac{e^{-t}}{2}+\left[\begin{array}{c}{-1}\\[4pt]{2}\\[4pt]{1}\end{array}\right]te^{-t} \right)\)
32. \({\bf y}=c_{1}\left[\begin{array}{c}{-1}\\[4pt]{1}\\[4pt]{0}\end{array}\right]e^{-2t}+c_{2}\left[\begin{array}{c}{0}\\[4pt]{0}\\[4pt]{1}\end{array}\right]e^{-2t}+c_{3}\left(\left[\begin{array}{c}{-1}\\[4pt]{0}\\[4pt]{0}\end{array}\right]e^{-2t}+\left[\begin{array}{c}{1}\\[4pt]{-1}\\[4pt]{1}\end{array}\right]te^{-2t} \right)\)